Re: Triangle with more than 180 degrees-

In article <1194564123.799213.297790@xxxxxxxxxxxxxxxxxxxxxxxxxxx> Hero <Hero.van.Jindelt@xxxxxx> writes:
*** wrote:
Hero wrote:
> *** wrote:
> > > This is application of Euclidian geometry too.
> > > Models are not realisations, here of a hyperbolic space.
> >
> > They are realisations. Moreover, they make it possible to visualise
> > how things work in hyperbolic space, and such give insight.
>
> Call them, whatever You like. But they are all done with the geometry
> of Euclid, so all four models have their shortcomings, f.e.they are
> done in a disk of finite radius, but the plane should have infinite
> size.

You read wrong. The metric is defined in terms of Euclidean metric, but
in the new metric the planes *have* infinite size. Size is related to the
metric used.

Using models, one has the metric of the model and the metric of the
realisation, or whatever You call it. Except for a model, which is one-
to-one, we have to calculate, f.e. 3 cm on a map of England might be
21 miles, when traveling through the landscape.

Yes, you have to calculate. You may note that a map is indeed a model of
the real thing, even if we disregard the landscape. But even when the
landscape is considered completely flat, there is no actual map such that
3 cm corresponds to 21 miles everywhere. The distortions when mapping
England are negligiable, but when you map the complete earth on a flat
map you will immediately see that it is impossible.

In hyperbolic space one has even an absolute measure, different from
the geometry of Euclid.

That is the point. There is no absolute measure. There is only the
measure you define.

In the geometry of Euclid, on the other hand,
one has similarity, which allows us to reduce a distance of 21 miles
to 3 cm of a model, but keeping angles and the numerical relations
between lengths, which is impossible in hyperbolic geometry.

You are wrong here. Take any model of hyperbolic geometry, and *use*
the metric defined on that model. Let's have a look at Klein's model.
He defines it on a disc x_1^2 + x_2^2 < 1. Let me define:
delta(x,y) = 1 - x_1.y_1 - x_2.y_2.
Then he defines a distance function as:
d(x,y) = a.arcosh[delta(x,y)/sqrt(delta(x,x).delta(y.y))]
See the constant a in front. Changing that constant changes distances,
but does not necessarily change angles, once you have defined angles.

Changing the size of a square in here will also change it's angles.

Well, that entirely depends on how you define angles.

Now, as hyperbolic geometry is non-euclidian, and therefore it starts
with a different axiom, one can not work with a concept of curvature,
which is defined upon the geometry of Euclid. And what is shown to us
as straight lines and as circles is obviously different.

They are different when you look at them from an Euclidean standpoint.

What you are bothering about is, I think, that when you live in a hyperbolic
space how it is possible that you experience only Euclidean space. The
point is that at a small scale all spaces are nearly Eucliden, the difference
is negligible. Your map of your favourite city will be Euclidean, and you
will find that it fits perfectly well with what you experience. Nevertheless
there are distortions, but they are negligible.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
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